If S = [sin (y – z) + sin (y + z) + 2 sin y] / [sin (x – z) + sin (x + z) + 2 sin x], then what is the simplified value of S?

Introduction / Context:
This question tests the use of standard trigonometric sum to product identities to simplify a complicated looking ratio. The expression is symmetric in structure for x and y, and contains sin of sum and difference along with extra sin terms. Recognizing this pattern is the key to a fast and clean simplification.

Given Data / Assumptions:

• S = [sin(y - z) + sin(y + z) + 2 sin y] / [sin(x - z) + sin(x + z) + 2 sin x].
• x, y, and z are real angles where all sines involved are defined.
• We are required to simplify S as much as possible.

Concept / Approach:
We use the identity sin(A + B) + sin(A - B) = 2 sin A cos B. This allows us to combine pairs of sine terms in both numerator and denominator. After applying the identity, both numerator and denominator share a common factor that can be cancelled, leaving a very simple ratio in terms of sin y and sin x.

Step-by-Step Solution:
Consider the numerator: sin(y - z) + sin(y + z). Using the identity sin(A + B) + sin(A - B) = 2 sin A cos B with A = y and B = z gives sin(y - z) + sin(y + z) = 2 sin y cos z. Therefore numerator = 2 sin y cos z + 2 sin y. Factor out 2 sin y: numerator = 2 sin y (cos z + 1). Now look at the denominator: sin(x - z) + sin(x + z). Again apply sin(A + B) + sin(A - B) = 2 sin A cos B with A = x and B = z to get sin(x - z) + sin(x + z) = 2 sin x cos z. Therefore denominator = 2 sin x cos z + 2 sin x = 2 sin x (cos z + 1). Now the ratio S becomes S = [2 sin y (cos z + 1)] / [2 sin x (cos z + 1)]. Cancel the common factor 2 (cos z + 1) to get S = sin y / sin x.

Verification / Alternative check:
We can verify by substituting particular values of x, y, and z that do not make the denominator zero, such as x = 60 degrees, y = 30 degrees, and z = 15 degrees. Evaluating both the original expression and sin y / sin x numerically produces the same value, confirming the simplification.

Why Other Options Are Wrong:
sin x tan y: This mixes sin y and sin x incorrectly and does not follow from the cancellations.
cos x sin y: This would arise if only some trigonometric factors were cancelled, leaving an extra cos x factor.
sin z: The final expression does not depend on z, since cos z + 1 cancels out from numerator and denominator.
cos y / cos x: Cosine does not appear in the final simplified expression after correct cancellation.

Common Pitfalls:
Errors often occur in applying the sum to product identity with the wrong A and B, or in forgetting to factor the common term 2 sin y or 2 sin x. Another error is cancelling only part of the common factor and leaving cos z or 2 behind, which leads to incorrect alternatives. Systematic application of identities and careful factoring avoid these issues.

Final Answer:
The simplified value of S is sin y / sin x.